I haven't watched the drama, (but it's on my list now) but is the trope you are looking for "forced cohabitation" or married for a green card?
There are two American movies, Green Card and The Proposal that play on green card marriages.
Marry me! Is a Japanese show, where the Government makes them get married
Shanai Marriage Honey is a sudden marriage between strangers.
Also Perfect and Casual, they get married so they can share a place if I remember correctly.